Generalized Forward-Backward Methods and Splitting Operators for a Sum of Maximal Monotone Operators

Suppose each of A(1),& mldr;, A(n) is a maximal monotone, and beta B is firmly nonexpansive with beta > 0. In this paper, we have two purposes: the first is finding the zeros of & sum;(n )(j=1)A(j )+ B, and the second is finding the zeros of & sum;(n )(j=1)A(j). To address the first problem, we produce fixed-point equations on the original Hilbert space as well as on the product space and find that these equations associate with crucial operators which are called generalized forward-backward splitting operators. To tackle the second problem, we point out that it can be reduced to a special instance of n = 2 by defining new operators on the product space. Iterative schemes are given, which produce convergent sequences and these sequences ultimately lead to solutions for the last two problems.